The Monty Hall Problem
You are on a game show in which there are three identical doors, one with a car behind it and two with goats behind them. You must pick one door, and you win if that door has the car behind it.
After you pick a door, the host of the game show always opens a door you didn’t choose that has a goat behind it. This leaves the door you chose and one other remaining door, and you are given the option to switch your choice to the other remaining door.
Should you switch or should you stick to your original choice? What chance of winning would that give you?
The History
The Monty Hall Problem is a classic probability puzzle, named for its similarity to the game show “Let’s Make a Deal”, which was hosted by Monty Hall. The problem was made famous when Marilyn vos Savant answered it correctly in her column in a popular magazine, and thousands of readers wrote letters to the magazine arguing her solution was wrong!
The solution can be counter-intuitive, so give it some thought and then scroll down to see the Monty Hall Problem explained.
The Solution
You should always switch to the other door, because it gives you a 2/3 chance of winning, as opposed to only a 1/3 chance of winning if you stick to your original choice.
Why is the probability of winning 2/3 instead of 1/2? Some might argue that you are left with two doors, and each is equally likely to have the car. However, the assumption that each is equally likely is incorrect, because you know the host always reveals a door you didn’t choose that has a goat behind it, which gives you information.
The Explanation
Different people find different explanations helpful, so here are a couple of different ways of looking at it:
Explanation 1
Since the 3 doors are identical, it is equally likely you pick the correct door, wrong door #1, or wrong door #2.
- 1/3 of the time you pick the correct door, and one of the other doors will be revealed (it doesn’t matter which). Switching means you lose.
- 1/3 of the time you pick wrong door #1, and wrong door #2 will be revealed. Switching means you win.
- 1/3 of the time you pick wrong door #2, and wrong door #1 will be revealed. Switching means you win.
We can see that you win by switching in 2 of the 3 equally likely scenarios.
Explanation 2
Your initial choice is correct 1/3 of the time, because you are choosing 1 out of 3 equally likely choices. Since you know you are always presented the option to switch, switching must therefore be correct the remainder of the time: 1 – 1/3 = 2/3.
Explanation 3
Let’s say instead of 3 doors, you have 100 doors. You randomly pick 1, and then the host reveals 98 doors with goats behind them, and asks if you want to switch. Obviously your original choice had a very small chance of being correct, and switching gives you a much better chance.
This explanation makes it easy to understand why switching is better, but it’s faulty because it glosses over the key insight. It’s not about how unlikely to be correct your original choice was, it’s about the information revealed when you know the host always reveals a wrong door.
The Key Insight
The way the problem is stated is important. The key insight is the host knows which door has the car, and always open a door with a goat behind it and offers you the choice to switch – this is the reason you can assume switching gives you the correct answer the remainder of the time.
Think of it this way: if the host only opens a door with a goat behind it if your original choice is correct, obviously you should never switch. So it’s not just the offer to switch that gives you the advantage, it’s knowing that the host always offers you the choice to switch.
Variations
Sometimes thinking through other scenarios helps you understand why the solution works under the original scenario. What if:
- The host only opens a door with a goat behind it if your original choice is correct.
- Switching is correct 0% of the time, because you only get the offer if your original choice was right.
- The host decides beforehand which door with a goat to reveal. After you pick a door, the host only opens the door with a goat if you did not pick the door that the host was going to reveal.
- Switching is correct 1/2 of the time. This is because 1/3 of the time you pick the right door and are given the option to switch, 1/3 of the time you pick the wrong door and are given the option to switch, and 1/3 of the time you pick the wrong door and are not given the option to switch. So you are given the option to switch in only 2 scenarios, and switching is correct in only 1 of those scenarios.
- After you pick a door, the host randomly picks one of the remaining doors, and only reveals it if a goat was behind it.
- Switching is correct 1/2 of the time. As above, 1/3 of the time you pick the right door and are given the option to switch, 1/3 of the time you pick the wrong door and are given the option to switch (because the host picked the remaining door with a goat behind it), and 1/3 of the time you pick the wrong door and are not given the option to switch (because the host picked the remaining door with the car behind it). Again, you are given the option to switch in only 2 scenarios, and switching is correct in only 1 of those scenarios.